Subgroups — #37555920

Order 23571.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14579756	C₃	1	1	yes	maximal, characteristic
2	#21463352	C₃	1	1	yes	maximal, characteristic, complemented
3	#21463351	C₃	1	3	yes	maximal, complemented
4	(C₉₇⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
5	(C₃×C₉₇⋊C₃)⋊C₉	C₃	1	3	yes	maximal
6	C₉×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₉⋊C₉×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₉₇⋊C₃	C₃²	1	1	yes	characteristic
9	#14579756	—	3	3	no	—
10	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
11	C₃×C₉₇⋊C₉	C₃²	1	3	yes	—
12	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
13	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₉₇	C₃²	1	3	yes	—
15	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
16	C₃²×C₉₇	C₃³	1	1	yes	characteristic
17	C₃×C₉₇⋊C₃	—	3	1	no	—
18	C₃×C₉₇⋊C₃	—	3	3	no	—
19	C₉×C₉₇	—	3	3	no	—
20	C₉×C₉₇	3₋¹⁺²	1	3	yes	—
21	C₉₇⋊C₉	—	3	1	no	—
22	C₉₇⋊C₉	—	3	1	no	—
23	C₉₇⋊C₉	—	3	3	no	—
24	C₉₇⋊C₉	—	3	3	no	—
25	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
26	C₃×C₉₇	#737	1	3	yes	—
27	C₉₇⋊C₃	—	9	1	no	—
28	#6899	—	97	1	no	maximal, Sylow
29	C₉₇	#6899	1	1	yes	characteristic, complemented, Sylow
30	C₃×3₋¹⁺²	—	97	1	no	—
31	C₉⋊C₉	—	97	3	no	—
32	C₉⋊C₉	—	97	3	no	—
33	C₉⋊C₉	—	97	1	no	—
34	C₉⋊C₉	C₉₇⋊C₃	1	1	yes	characteristic, complemented
35	C₃²⋊C₉	—	97	3	no	—
36	C₉²	—	97	1	no	—
37	C₃³	—	97	1	no	—
38	3₋¹⁺²	—	291	3	no	—
39	C₃×C₉	—	97	3	no	—
40	C₃×C₉	—	97	3	no	—
41	C₃×C₉	—	97	1	no	—
42	C₃×C₉	—	97	1	no	—
43	C₃×C₉	C₃×C₉₇⋊C₃	1	3	yes	—
44	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
45	C₃²	—	291	1	no	—
46	C₃²	—	291	3	no	—
47	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
48	C₉	—	3	3	no	—
49	C₉	—	291	1	no	—
50	C₉	—	291	1	no	—
51	C₉	—	291	3	no	—
52	C₉	—	291	3	no	—
53	C₉	#14579756	1	3	yes	—
54	C₃	—	873	1	no	—
55	C₃	C₃×#14579756	1	1	yes	characteristic
56	C₃	#21463378	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₉₇⋊C₃
commutator, socle, cosocle	C₃²×C₉₇	C₃³
Fitting	C₉⋊C₉×C₉₇	C₃
prime radical	C₃²×C₉₇⋊C₃	C₃²
hypercenter	C₉⋊C₉	C₉₇⋊C₃
hypocenter	C₉₇	#6899
LCS	G ▷ C₃²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #21463352	C₃	semidirect
3. #21463351	C₃	semidirect
4. (C₉₇⋊C₉)⋊C₉	C₃	semidirect
6. C₉×C₉₇⋊C₉	C₃	semidirect
7. C₉⋊C₉×C₉₇	C₃	semidirect
29. C₉₇	#6899	semidirect
34. C₉⋊C₉	C₉₇⋊C₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#14579756	3
#21463352	3
#21463351	3
(C₉₇⋊C₉)⋊C₉	3
(C₃×C₉₇⋊C₃)⋊C₉	3
C₉×C₉₇⋊C₉	3
C₉⋊C₉×C₉₇	3
#6899	97